normal_scores_approxcalculates an approximation to the set of Normal Scores, i.e., the expected values of an ordered set of independent observations from a Normal distribution with mean and standard deviation .
For full information please refer to the NAG Library document for g01db
, the size of the sample.
- ppfloat, ndarray, shape
The Normal scores. contains the value , for .
- (errno )
On entry, .
No equivalent traditional C interface for this routine exists in the NAG Library.
normal_scores_approxis an adaptation of the Applied Statistics Algorithm AS , see Royston (1982). If you are particularly concerned with the accuracy with which
normal_scores_approxcomputes the expected values of the order statistics (see Accuracy), then
normal_scores_exact()which is more accurate should be used instead at a cost of increased storage and computing time.
Let be the order statistics from a random sample of size from the standard Normal distribution. Defining
where is the expected value of , the current function approximates the Normal upper tail area corresponding to as,
, for , and . Estimates of , , , and are obtained. A small correction to is necessary when and .
The approximation to is thus given by
Values of the inverse Normal probability integral are obtained from
Royston, J P, 1982, Algorithm AS 177: expected normal order statistics (exact and approximate), Appl. Statist. (31), 161–165